The students will work in small groups on an
activity. We will take notes together as a class to show examples
of linear systems with one solution, no solution, or infinitely many
solutions.

Prerequisites:

Before this lesson
students should know what slope and intercepts are. Students should
know how to put equations in slope-intercept form, how to graph
linear equations, and methods for solving systems of equations.

Standards:

9-12.A.2.1. Students
are able to use algebraic properties to transform multi-step,
single-variable, first-degree equations.

9-12.A.2.2A.
Students are able to determine the solution of systems of equations
and systems of inequalities.

9-12.A.3.1. Students are able to create linear models to represent
problem situations.

9-12.A.4.1. Students are able to use graphs, tables, and equations
to represent linear functions.

9-12.N.2.1. Students are able to add, subtract,
multiply, and divide real numbers including integral exponents.

Lesson Objectives:

Identify linear
systems as having one solution, no solution, or infinitely many
solutions

Assessment:

Assignment: pages
429-431 problems 12-28, 36-37. Quiz (to be taken on day following
lesson)

Technology to be Used:

Overhead Projector

Other Materials:

Graph paper;
whiteboard

Procedural Activities:

Lesson Opener:
Interpersonal intelligence is used during this activity when the
students work in small groups and partake in group discussion.
Because students will be solving and graphing the equations
themselves, they will be learning by doing therefore demonstrating
bodily-kinesthetic intelligence during the activity.

Give each student a
piece of graph paper. Divide students into groups of three at their
desks. Give the students the directions for the activity as
follows:

Each member of the
group should choose a different one of the linear equations that
will be on the overhead and graph it.

Rotate papers once to
your left and check the other students work and graph.

Everyone should get
their own papers back and share your graphs with the other
members. How are the three graphs different? (How many lines
does each graph have? If there is more than one line, how are
the lines related?)

Individually, write
the equations of your system in the form y=mx+b.

Share with your group
member what you got for a result. How are the equations within
each system similar or different?

Repeat the same steps
with the second set of three systems of equations.

We will now have a
large group discussion to see what the groups discovered from the
activity. From this activity students should discover the three
following ideas:

One solution if
each of the equations have different slopes.

No solution if
the lines are parallel (same slopes and different

y-intercepts).

Infinitely many
solutions if the equations are of the same line (same slopes
and same y-intercepts).

Lecture/Notes:Verbal-linguistic intelligence is
being used during lecture because students will be hearing all the
important points and may follow along in the book with vocabulary.
They may also refer to examples in the book for further reading.
Along with this, students should be taking notes as we work problems
out on the board.

Systems of equations
can have either one solution, no solution, or many solutions.

The number of
solutions a system has can be determined from the graph or from
the equations.

In
each of the following examples, when doing the “solve by graphing”
all students will be doing the graphing on their graphing
calculators. Because they will be learning by doing they are using
bodily-kinesthetic intelligence. They are also using visual-spatial
intelligence because they are interpreting graphs and determining
relationships between lines on the graph.

Each
example will be worked out step-by-step on the board during
lecture. Each steps shows logical progression toward the answer,
thus logical-mathematical intelligence will be used.
Following are the examples I will use in class, each step is not
written here, but will be written for students notes during class.

Solve
by graphing: appears like one line because they are the same line

Example #4
Identifying the Number of Solutions

6x – 2y = 10

-3x + y = 12

Solve
by substitution: 24 = 10 false so no solution.

What
would the graph look like? Parallel lines.

Check
by graphing on calculators.

Closing the
lesson: I will reemphasize
how we can predict what a graph will look like by comparing the
equations. Students will then be given their homework assignment
and have five minutes to get started on it or ask questions.